Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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Can we define flat connection on any given smooth manifold?

For example, a sphere $S^2$ in $\mathbf{R}^3$ is apparently not flat with respect to the Euclidean connection, but can we define a flat connection and thus with affine charts on $S^2$?
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50 views

Is $T\mathbb{C}\mathbb{P}^n$ globally generated?

A vector bundle $E\to X$ is globally generated if there exists global holomorphic sections $s_1,\dots,s_n$ such that $E_x$ is spanned by $s_1(x),\dots,s_n(x)$ for all $x\in X$. Consider the ...
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12 views

stereographic projection, coordinates

when you want to map a random point on the sphere $S^n\subset\mathbb{R}^{n+1}$. And observe the points $e_\pm=(0,\dotso,0,\pm 1)$, you can use the function $g_+(t)=e_++t(x_1,\dotso, x_n,x_{n+1}-1)$ ...
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100 views

Is Floer homology always isomorphic to the singular homology of some space?

After I studied Morse homology, I'm now studying the following Floer homology theories : 1) Symplectic Floer homology ; 2) Floer homology of lagrangians ; 3) Heegard-Floer homology ; ...
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19 views

Commuting derivatives in a Lie group

Let $G$ be a Lie group and $f : \mathbb{R}^2 \to G$ smooth. Consider $\theta \in \Omega^1(G, \mathfrak{g})$ the Maurer-Cartan form of $G$. I'm trying to understand why $\displaystyle\frac{d}{dt} \...
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1answer
25 views

A book on Vector Calculus with emphasis on geometrical intuition

I am a physicist trying to learn vector calculus in a way that is a mixture of the way mathematicians learn it with the way that physicist learn it in order to be able to learn Differential Geometry ...
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2answers
27 views

convention of a default atlas

Recently, I have been studying the basics of differential geometry and te necessary preliminaries. I arrived at the construction of differential structure on topological manifolds, where the non-...
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29 views

smooth manifolds, equivalent statements

Let $X,Y$ be smooth manifolds. Show: A function $f:X\to Y$ is smooth, iff for every open $V\subseteq Y$ and every smooth function $g:V\to\mathbb{R}$ the composition $g\circ f: f^{-1}(V)\to\mathbb{R}$ ...
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23 views

covering space, smooth manifold

Let $p:Y\to X$ be a covering space and $p^{-1}(x)$ countable for every $x\in X$. Task: Let $X$ be a smooth manifold. Show, that $Y$ has the structure of a smooth manifold, regarding this $p$ is ...
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1answer
26 views

Rotaion Surfaces and Complex Numbers

Consider a continuous invertible map $\varphi:\mathbb{R}^+ \longrightarrow \mathbb{R}$, and define the follwing surface $$ s:\mathbb{C} \longrightarrow \mathbb{R} \times \mathbb{C} $$ $$ \qquad xe^{i\...
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178 views

Is this contraction of metric tensor derivatives symmetric?

A couple of times when I've tried to prove symmetries of various tensors (for learning), I've ended up with the expression below, and the fact that either a) I made mistake, or b) the expression is ...
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29 views

Incompatible first and second fundamental forms

Say the first and second fundamental forms of a surface (a and b) in 2D are incompatible (i.e. they do not satisfy the Codazzi-Mainardi equations), then the "surface" cannot be embedded in 3D. Is this ...
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58 views

Is the contraction of an harmonic form harmonic?

As I'm still a beginner in complex differential geometry, as soon as I tried reading an article I got stuck on what (I think) should be a minor detail. I hope someone can help me a bit. Let $M$ be a ...
4
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1answer
104 views

Diffeomorphisms between factors in diffeomorphic product manifolds

Let $M$, $N$ and $P$ be three smooth manifolds such that $M \times N$ is diffeomorphic to $M \times P$. I need to know about some conditions under which one can deduce that $N$ is diffeomorphic to $P$....
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33 views

Smooth structure on open subsets of manifolds

Let $X$ be a smooth manifold and $U\subseteq X$ open. Define a canonical smooth structure on $U$, for which the embedding $U\to X$ is smooth. Hello, I want to solve this task. My try was as follows:...
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23 views

Equation of silhouette from an arbitrary viewpoint

A two parameter $(u,v)$ surface in $\mathbb R^3$ when viewed from a point at infinite distance casts a shadow on any given plane. What ODE/PDE describes its envelope of its silhouetted projection? ...
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59 views

Does the associated bundle functor have left or right adjoints?

Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and $\bar{\...
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57 views

Automorphisms action on $\mathbb C$

If we know that the $$Aut \ \mathbb C=\{z\mapsto az+b:a\in \mathbb C^*,b\in \mathbb C\}$$ What are the subgroups of $Aut \ \mathbb C$ that act withiut fixed points and properly discontinuously on $\...
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24 views

The action of automorphisms on the Riemann sphere

If we are given that the automorphism group of the Riemann sphere is $$Aut\ \mathbb P^1=\{z\mapsto \frac{az+b}{cz+d}:ad-bc=1\}$$ Why this group does not have any proper subgroups that act without ...
3
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2answers
71 views

Defining a Riemannian metric

Let $M$ be a smooth manifold. I have seen a Riemannian metric be defined in many ways: A smooth choice of an inner product $g_p:T_pM\times T_pM\to\mathbb{R}$ which is symmetric and positive-definite,...
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59 views

Exponential of Lie Groups.

When the exponential map defined a bijection between the group G and their Lie algebra? The only example I know is the Heisenberg group.
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Is $T_pM$ isomorphic to $T_{F(p)}N$?

Let $M$ and $N$ be two smooth n-dimensional manifolds and $F:M\to N$ be a diffeomorphism. Is it true that $F_{*p}:T_pM\longrightarrow T_{F(p)}N$ is an isomorphism?
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Ratio of holomorphic forms on a Riemann surface

Let $R$ be a Riemann surface of genus $g>1$ and $\omega$, $\sigma$ two holomorphic 1-forms on $R$. This means that locally we can write $\omega=fdz$ and $\sigma=gdz$ with $f$ and $g$ holomorphic. ...
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Coordinates of exterior derivative of dual basis of local frame for the tangent bundle

Let $M$ be an $n$-manifold. Let $E_1, E_2,\dots, E_n : U\subset M \to TM $ be a local frame for $TM$ with associated local dual frame $\epsilon^1, \epsilon^2,\dots, \epsilon^n : U\subset M \to T^*M $. ...
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1answer
17 views

extending functions from the horizontal bundle to the whole bundle

Let $(M,g)$ be a Riemannian manifold and $G$ a compact Lie group acting freely and isometrically on $M$. Let $\pi \colon M \to M/G$ be the projection to the orbits. Using the metric, we get a ...
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Quick question: Curvature form of a connection on the trivial bundle

Let $L=\mathbb{R}^2\times U(1)$ be the trivial $U(1)$-bundle over $\mathbb{R}^2$. Define a connection $\nabla=d+A$ where $A=fdx+gdy$ is an $\mathbb{R}$ valued $1$-form on $L$. That is, $\nabla$ gives ...
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1answer
63 views

How to build smooth variations along arbitrary paths at the endpoint?

Let $M$ be a smooth manifold, $\gamma:[0,a] \to M$ a smooth path. Assume we are given another smooth path $\phi:(-\epsilon,\epsilon) \to M$ that starts from an endpoint of $\gamma$, i.e $\phi(0)=\...
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2answers
51 views

Differential of the multiplication and inverse maps on a Lie group

I'm trying to solve the following two problems: Let $G$ be a Lie group with multiplication map $\mu\colon G\times G\to G$, and let $\ell_a\colon G\to G$ and $r_b\colon G\to G$ be left and right ...
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25 views

Is this true or false? Tangent Space isomorphism [on hold]

Is this a consequence of the third isomorphism theorem? Let $L \subset N \subset G$, then $T(G/N) \simeq T(G/N) / T(N/L) $?
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Existence of submersions between manifolds

I have a ton of problems, where I need to prove (or disprove) the existence of submersions between given manifolds. I will give you some examples, and hopefully I can learn the techniques to solve all ...
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Existence of map between projective planes.

I'm struggling to solve this problem, any indications would be appreciated. Is there an application $f : \mathbb RP^3 \to \mathbb RP^1$ of class $\mathcal C^3$ such that $f^{-1}(p)$ is the union of ...
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Derivative of projection

Let $\Omega \subset \mathbb{R}^n$ be a limited domain of class $C^\infty$ (open, connected) and $\varepsilon > 0$ small such that $$ \Omega_\varepsilon = \{y \in \overline{\Omega} : d(y, \partial \...
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Need help understanding part of this proof about local coordinates for Legendrian manifold

I need help understanding this proof in this book here: Concretely, I do not understand why it is okay to assume that $S$ can be parameterized by $n$ variables. Sure, it's an $n$-dimensional ...
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22 views

Roadmap to Differential Geometry for Machine Learning

Recently within machine learning, there are a lot of works on non-convex optimization and natural gradients methods etc which are based on differential geometry, it gives rise to increased need to ...
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1answer
77 views

Which concepts in Differential Geometry can NOT be represented using Geometric Algebra?

1. It is not clear to me that linear duals, and not just Hodge duals, can be represented in geometric algebra at all. See, for example, here. Can linear duals (i.e. linear functionals) be ...
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1answer
20 views

Definition of formal adjoint of covariant derivative

I read in Einstein Manifolds, L. Besse that the covariant derivative $D: \mathcal{J}^{(r,s)}(M)\to \Omega^1(M)\otimes\mathcal{J}^{(r,s)}(M)$ admit an formal adjoint $D^*:\Omega^1(M)\otimes\mathcal{J}^{...
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Clarification for Manifold and Manifold with Boundary Definitions

I am reading Spivak's Calculus on Manifolds, and just need a bit of clarification when it comes to definitions. He defines a manifold as some space $M$ that satisfies: $(M)$: $\forall x \in M, \...
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1answer
28 views

Compute Christoffel symbols of a rotating cartesian coordinate system

Suppose we have a smooth manifold $(\mathbb{R}^3, \mathcal{O}_{\mathbb{R}^4}, \{(\mathbb{R}^3,x),(\mathbb{R}^3,y)\},\nabla,t)$ where $t:\mathbb{R}^3\rightarrow\mathbb{R}$ is such that $t(a,b,c,d)=a$, $...
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Unique Solution to Equation in Two Variables & Possible Use of the Implicit Function Theorem

Let $g(x) : R \to R$ be a continuous function; Consider the equation $ T(x,y) = y^3 -y^2 +(1+x^2)y - g(x)$ Show that for a given $x$ there exist a unique solution $y$ to the equation $T(x,y) = 0$. ...
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1answer
29 views

Why is the Christoffel symbol of the 2nd kind symmetric in lower indices?

I have consulted multiple books on tensors for physicists, but they all take for granted this relation: $\Gamma_{ij}^k = \Gamma_{ji}^k$ However, no proof is provided and I cannot find a single one ...
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36 views

Function as a combination of 1-forms on a Riemann surface

My question is quite simple, I hope it's not also stupid.. Consider $R$ a Riemann surface and $\omega_1$, $\omega_2$ two $(1,0)$-forms (i.e. holomorphic forms) and $\varphi_1$, $\varphi_2$ two $(0,1)$-...
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Given parametrization of torus is equal to level surface

I need to show that the torus $T=\pmatrix{(R+r\cos\phi)\cos \theta\\(R+r\cos\phi)\sin\theta\\r\sin\phi}$ is equal to the surface given implicitly by $(\sqrt{x^2+y^2}-R)^2+z^2-r^2=0$. I already got ...
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Reference about the surgery of Ricci flow

I roughly read the Topping's LECTURES ON THE RICCI FLOW. Seemly, there is not introduction about surgery. Seemly,it is enough to deal singularity by blow up. Then, for knowing surgery I read the ...
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Need help understanding local coordinates of differential forms

I was trying to check a variation of theorem 2.2: I did the first part of the proof using the standard contact structure on $\mathbb R^{2n + 1}$ which worked out as expected. Then I wanted to do it ...
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38 views

Pullback Courant algebroid

I am reading the lecture notes on generalized geometry http://www.staff.science.uu.nl/~caval101/homepage/Research_files/australia.pdf but now I am stuck on the exercise 1.31 at page 13. Here is the ...
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What is the torsion and curvature in modern language?

I think it can not be the torsion and curvature in the connection context, for these two are anti-symmetric in two variables, so that they must vanish on a one-dimensional space (tangent space of a ...
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1answer
43 views

Is a surjective mapping of R2 to itself with full rank derivative everywhere necessarily injective?

If $f:\mathbb R^2\rightarrow\mathbb R^2$ has rank 2 derivative everywhere, then by the inverse function theorem it is locally injective. If it is surjective, is it then necessarily globally injective ...
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128 views

Envelope of family of curves $x(u,v)=\cos^2(u)\cos(v)+\cos(u)\sin(u)\sin(v)$, $y(u,v)=\cos^2(u)\sin(v)-\cos (u)\sin(u)\cos(v)$

How to generally find singular solution or envelope of a two parameter family of curves $ x(u,v),y(u,v) $ in the plane? The parametric equations $$x(u,v) = \cos^2 (u) \cos (v) + \cos (u) \sin (u) \...
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26 views

Smooth structure in reconstruction theorem

Let $M,F$ be smooth manifolds, $\{U_i:i\in I\}$ an open cover of $M$ and a cocycle $\{t_{ij}:U_i\cap U_j\to\mathrm{Diff}(F)\}$. In almost any book which discusses fibre bundles, one can find the ...
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1answer
25 views

Curve in a disk

I take a curve $\vec\gamma:[a,b]\longrightarrow \Delta$, where $\Delta \subseteq \mathbb{R}^2$ is the disk of radius $r>0$. If the curve has length $L>0$ does exist an upper bound (in terms of $...